How to explain it
The anchor students hold onto: A compound event occurs when two or more simple events happen together. P(compound event) = number of favorable outcomes / total outcomes in the sample space.
You can now find probabilities for compound events by building sample spaces. This thinking connects to statistical reasoning and simulation — tools you will use in 8th grade data analysis.
Worked examples
Example 1
Organized List
Flip a coin and roll 1-3.
Step 1Sample space: (H,1)(H,2)(H,3)(T,1)(T,2)(T,3) — 6 outcomes.
Step 2Favorable (H and 2): 1 outcome.
Step 3P(heads and 2) = 1/6.
AnswerP(heads and 2) = 1/6
Example 2
Tree Diagram
Spinner (R/B) and coin flip.
Step 1Tree: (R,H)(R,T)(B,H)(B,T) — 4 outcomes.
Step 2Favorable (Red and Heads): 1 outcome.
Step 3P(Red and Heads) = 1/4.
AnswerP(Red and Heads) = 1/4
Example 3
Without Replacement
Pick 2 tiles; no replacement.
Step 1Bag: Red, Blue, Yellow. Draw 2 without replacing.
Step 2Outcomes: (R,B)(R,Y)(B,R)(B,Y)(Y,R)(Y,B) — 6 total.
Step 3P(Red, then Blue) = 1/6.
AnswerP(Red then Blue) = 1/6
Common mistakes
What students write
Builds an incomplete tree diagram or organized list, missing some outcome pairs, and uses too few total outcomes to compute P.
The fix
Systematically list every combination: pair each first-event outcome with every second-event outcome. Total outcomes = (# first outcomes) x (# second outcomes). Count all paths before computing P.
Try this
Rivera uses a tree diagram to find the probability of drawing a Red tile and then a Blue tile from a bag of 3 tiles (Red, Blue, Yellow), drawn without replacement. Rivera's work: Tree: Red — Blue — Yellow (3 outcomes total) P(Red, then Blue) = 1/3 Identify Rivera's error and find the correct probability.
What students write
Treats a without-replacement (dependent) draw as independent, using the original full count for the second draw and getting a sample space that is too large.
The fix
When items are drawn without replacement, reduce the available outcomes for the second draw by 1. List the new sample space: fewer outcomes are available after the first item is removed.
Try this
Tran draws 2 tiles from a bag containing 4 tiles (Red, Blue, Green, Yellow) without replacement and finds P(Red first, then Blue). Tran's work: The bag has 4 tiles, so the sample space has 4 x 4 = 16 outcomes. P(Red first, then Blue) = 1/16 Identify Tran's error and find the correct probability.
Teacher tip
Head off the two predictable errors before they happen. First: Systematically list every combination: pair each first-event outcome with every second-event outcome. Total outcomes = (# first outcomes) x (# second outcomes). Count all paths before computing P. Second: When items are drawn without replacement, reduce the available outcomes for the second draw by 1. List the new sample space: fewer outcomes are available after the first item is removed.